= Gauge theory =

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= Gauge theory =
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Introduction
======================================================================
In physics, a gauge theory is a type of field theory in which the
Lagrangian is invariant under certain Lie groups of local
transformations.

The term 'gauge' refers to any specific mathematical formalism to
regulate redundant degrees of freedom in the Lagrangian. The
transformations between possible gauges, called 'gauge
transformations', form a Lie group�referred to as the 'symmetry group'
or the 'gauge group' of the theory. Associated with any Lie group is
the Lie algebra of group generators. For each group generator there
necessarily arises a corresponding field (usually a vector field)
called the 'gauge field'. Gauge fields are included in the Lagrangian
to ensure its invariance under the local group transformations (called
'gauge invariance'). When such a theory is quantized, the quanta of
the gauge fields are called 'gauge bosons'. If the symmetry group is
non-commutative, then the gauge theory is referred to as non-abelian
gauge theory, the usual example being the Yang-Mills theory.

Many powerful theories in physics are described by Lagrangians that
are invariant under some symmetry transformation groups. When they are
invariant under a transformation identically performed at 'every'
point in the spacetime in which the physical processes occur, they are
said to have a global symmetry. Local symmetry, the cornerstone of
gauge theories, is a stronger constraint. In fact, a global symmetry
is just a local symmetry whose group's parameters are fixed in
spacetime (the same way a constant value can be understood as a
function of a certain parameter, the output of which is always the
same).

Gauge theories are important as the successful field theories
explaining the dynamics of elementary particles. Quantum
electrodynamics is an abelian gauge theory with the symmetry group
U(1) and has one gauge field, the electromagnetic four-potential, with
the photon being the gauge boson. The Standard Model is a non-abelian
gauge theory with the symmetry group U(1) � SU(2) � SU(3) and has a
total of twelve gauge bosons: the photon, three weak bosons and eight
gluons.

Gauge theories are also important in explaining gravitation in the
theory of general relativity. Its case is somewhat unusual in that
the gauge field is a tensor, the Lanczos tensor. Theories of quantum
gravity, beginning with gauge gravitation theory, also postulate the
existence of a gauge boson known as the graviton. Gauge symmetries
can be viewed as analogues of the principle of general covariance of
general relativity in which the coordinate system can be chosen freely
under arbitrary diffeomorphisms of spacetime. Both gauge invariance
and diffeomorphism invariance reflect a redundancy in the description
of the system. An alternative theory of gravitation, gauge theory
gravity, replaces the principle of general covariance with a true
gauge principle with new gauge fields.

Historically, these ideas were first stated in the context of
classical electromagnetism and later in general relativity. However,
the modern importance of gauge symmetries appeared first in the
relativistic quantum mechanics of electronsquantum electrodynamics,
elaborated on below. Today, gauge theories are useful in condensed
matter, nuclear and high energy physics among other subfields.

History
======================================================================
The earliest field theory having a gauge symmetry was Maxwell's
formulation, in 1864-65, of electrodynamics ("A Dynamical Theory of
the Electromagnetic Field") which stated that any vector field whose
curl vanishes�and can therefore normally be written as a gradient of a
function�could be added to the vector potential without affecting the
magnetic field. The importance of this symmetry remained unnoticed in
the earliest formulations. Similarly unnoticed, Hilbert had derived
the Einstein field equations by postulating the invariance of the
action under a general coordinate transformation. Later Hermann Weyl,
in an attempt to unify general relativity and electromagnetism,
conjectured that 'Eichinvarianz' or invariance under the change of
scale (or "gauge") might also be a local symmetry of general
relativity. After the development of quantum mechanics, Weyl,
Vladimir Fock and Fritz London modified gauge by replacing the scale
factor with a complex quantity and turned the scale transformation
into a change of phase, which is a U(1) gauge symmetry. This explained
the electromagnetic field effect on the wave function of a charged
quantum mechanical particle. This was the first widely recognised
gauge theory, popularised by Pauli in 1941.

In 1954, attempting to resolve some of the great confusion in
elementary particle physics, Chen Ning Yang and Robert Mills
introduced non-abelian gauge theories as models to understand the
strong interaction holding together nucleons in atomic nuclei. (Ronald
Shaw, working under Abdus Salam, independently introduced the same
notion in his doctoral thesis.) Generalizing the gauge invariance of
electromagnetism, they attempted to construct a theory based on the
action of the (non-abelian) SU(2) symmetry group on the isospin
doublet of protons and neutrons. This is similar to the action of the
U(1) group on the spinor fields of quantum electrodynamics. In
particle physics the emphasis was on using quantized gauge theories.

This idea later found application in the quantum field theory of the
weak force, and its unification with electromagnetism in the
electroweak theory. Gauge theories became even more attractive when it
was realized that non-abelian gauge theories reproduced a feature
called asymptotic freedom. Asymptotic freedom was believed to be an
important characteristic of strong interactions. This motivated
searching for a strong force gauge theory. This theory, now known as
quantum chromodynamics, is a gauge theory with the action of the SU(3)
group on the color triplet of quarks. The Standard Model unifies the
description of electromagnetism, weak interactions and strong
interactions in the language of gauge theory.

In the 1970s, Michael Atiyah began studying the mathematics of
solutions to the classical Yang-Mills equations. In 1983, Atiyah's
student Simon Donaldson built on this work to show that the
differentiable classification of smooth 4-manifolds is very different
from their classification up to homeomorphism. Michael Freedman used
Donaldson's work to exhibit exotic R4s, that is, exotic differentiable
structures on Euclidean 4-dimensional space. This led to an increasing
interest in gauge theory for its own sake, independent of its
successes in fundamental physics. In 1994, Edward Witten and Nathan
Seiberg invented gauge-theoretic techniques based on supersymmetry
that enabled the calculation of certain topological invariants (the
Seiberg-Witten invariants). These contributions to mathematics from
gauge theory have led to a renewed interest in this area.

The importance of gauge theories in physics is exemplified in the
tremendous success of the mathematical formalism in providing a
unified framework to describe the quantum field theories of
electromagnetism, the weak force and the strong force. This theory,
known as the Standard Model, accurately describes experimental
predictions regarding three of the four fundamental forces of nature,
and is a gauge theory with the gauge group SU(3) � SU(2) � U(1).
Modern theories like string theory, as well as general relativity,
are, in one way or another, gauge theories.

:'See Pickering
for more about the history of gauge and quantum field theories.'

Global symmetry
=================
In physics, the mathematical description of any physical situation
usually contains excess degrees of freedom; the same physical
situation is equally well described by many equivalent mathematical
configurations. For instance, in Newtonian dynamics, if two
configurations are related by a Galilean transformation (an inertial
change of reference frame) they represent the same physical situation.
These transformations form a group of "symmetries" of the theory, and
a physical situation corresponds not to an individual mathematical
configuration but to a class of configurations related to one another
by this symmetry group.

This idea can be generalized to include local as well as global
symmetries, analogous to much more abstract "changes of coordinates"
in a situation where there is no preferred "inertial" coordinate
system that covers the entire physical system. A gauge theory is a
mathematical model that has symmetries of this kind, together with a
set of techniques for making physical predictions consistent with the
symmetries of the model.

Example of global symmetry
============================
When a quantity occurring in the mathematical configuration is not
just a number but has some geometrical significance, such as a
velocity or an axis of rotation, its representation as numbers
arranged in a vector or matrix is also changed by a coordinate
transformation. For instance, if one description of a pattern of
fluid flow states that the fluid velocity in the neighborhood of
('x'=1, 'y'=0) is 1 m/s in the positive 'x' direction, then a
description of the same situation in which the coordinate system has
been rotated clockwise by 90 degrees states that the fluid velocity in
the neighborhood of ('x'=0, 'y'=1) is 1 m/s in the positive 'y'
direction. The coordinate transformation has affected both the
coordinate system used to identify the 'location' of the measurement
and the basis in which its 'value' is expressed. As long as this
transformation is performed globally (affecting the coordinate basis
in the same way at every point), the effect on values that represent
the 'rate of change' of some quantity along some path in space and
time as it passes through point 'P' is the same as the effect on
values that are truly local to 'P'.

Use of fiber bundles to describe local symmetries
===================================================
In order to adequately describe physical situations in more complex
theories, it is often necessary to introduce a "coordinate basis" for
some of the objects of the theory that do not have this simple
relationship to the coordinates used to label points in space and
time. (In mathematical terms, the theory involves a fiber bundle in
which the fiber at each point of the base space consists of possible
coordinate bases for use when describing the values of objects at that
point.) In order to spell out a mathematical configuration, one must
choose a particular coordinate basis at each point (a 'local section'
of the fiber bundle) and express the values of the objects of the
theory (usually "fields" in the physicist's sense) using this basis.
Two such mathematical configurations are equivalent (describe the same
physical situation) if they are related by a transformation of this
abstract coordinate basis (a change of local section, or 'gauge
transformation').

In most gauge theories, the set of possible transformations of the
abstract gauge basis at an individual point in space and time is a
finite-dimensional Lie group. The simplest such group is U(1), which
appears in the modern formulation of quantum electrodynamics (QED) via
its use of complex numbers. QED is generally regarded as the first,
and simplest, physical gauge theory. The set of possible gauge
transformations of the entire configuration of a given gauge theory
also forms a group, the 'gauge group' of the theory. An element of the
gauge group can be parameterized by a smoothly varying function from
the points of spacetime to the (finite-dimensional) Lie group, such
that the value of the function and its derivatives at each point
represents the action of the gauge transformation on the fiber over
that point.

A gauge transformation with constant parameter at every point in space
and time is analogous to a rigid rotation of the geometric coordinate
system; it represents a global symmetry of the gauge representation.
As in the case of a rigid rotation, this gauge transformation affects
expressions that represent the rate of change along a path of some
gauge-dependent quantity in the same way as those that represent a
truly local quantity. A gauge transformation whose parameter is 'not'
a constant function is referred to as a local symmetry; its effect on
expressions that involve a derivative is qualitatively different from
that on expressions that don't. (This is analogous to a non-inertial
change of reference frame, which can produce a Coriolis effect.)

Gauge fields
==============
The "gauge covariant" version of a gauge theory accounts for this
effect by introducing a 'gauge field' (in mathematical language, an
Ehresmann connection) and formulating all rates of change in terms of
the covariant derivative with respect to this connection. The gauge
field becomes an essential part of the description of a mathematical
configuration. A configuration in which the gauge field can be
eliminated by a gauge transformation has the property that its field
strength (in mathematical language, its curvature) is zero everywhere;
a gauge theory is 'not' limited to these configurations. In other
words, the distinguishing characteristic of a gauge theory is that the
gauge field does not merely compensate for a poor choice of coordinate
system; there is generally no gauge transformation that makes the
gauge field vanish.

When analyzing the dynamics of a gauge theory, the gauge field must be
treated as a dynamical variable, similar to other objects in the
description of a physical situation. In addition to its interaction
with other objects via the covariant derivative, the gauge field
typically contributes energy in the form of a "self-energy" term. One
can obtain the equations for the gauge theory by:
* starting from a naïve ansatz without the gauge field (in which the
derivatives appear in a "bare" form);
* listing those global symmetries of the theory that can be
characterized by a continuous parameter (generally an abstract
equivalent of a rotation angle);
* computing the correction terms that result from allowing the
symmetry parameter to vary from place to place; and
* reinterpreting these correction terms as couplings to one or more
gauge fields, and giving these fields appropriate self-energy terms
and dynamical behavior.
This is the sense in which a gauge theory "extends" a global symmetry
to a local symmetry, and closely resembles the historical development
of the gauge theory of gravity known as general relativity.

Physical experiments
======================
Gauge theories used to model the results of physical experiments
engage in:
* limiting the universe of possible configurations to those consistent
with the information used to set up the experiment, and then
* computing the probability distribution of the possible outcomes that
the experiment is designed to measure.
We cannot express the mathematical descriptions of the "setup
information" and the "possible measurement outcomes", or the "boundary
conditions" of the experiment, without reference to a particular
coordinate system, including a choice of gauge. One assumes an
adequate experiment isolated from "external" influence that is itself
a gauge-dependent statement. Mishandling gauge dependence
calculations in boundary conditions is a frequent source of anomalies,
and approaches to anomaly avoidance classifies gauge theories.

Continuum theories
====================
The two gauge theories mentioned above, continuum electrodynamics and
general relativity, are continuum field theories. The techniques of
calculation in a continuum theory implicitly assume that:
* given a completely fixed choice of gauge, the boundary conditions of
an individual configuration are completely described
* given a completely fixed gauge and a complete set of boundary
conditions, the least action determines a unique mathematical
configuration and therefore a unique physical situation consistent
with these bounds
* fixing the gauge introduces no anomalies in the calculation, due
either to gauge dependence in describing partial information about
boundary conditions or to incompleteness of the theory.
Determination of the likelihood of possible measurement outcomes
proceed by:
* establishing a probability distribution over all physical situations
determined by boundary conditions consistent with the setup
information
* establishing a probability distribution of measurement outcomes for
each possible physical situation
* convolving these two probability distributions to get a distribution
of possible measurement outcomes consistent with the setup information
These assumptions have enough validity across a wide range of energy
scales and experimental conditions to allow these theories to make
accurate predictions about almost all of the phenomena encountered in
daily life: light, heat, and electricity, eclipses, spaceflight, etc.
They fail only at the smallest and largest scales due to omissions in
the theories themselves, and when the mathematical techniques
themselves break down, most notably in the case of turbulence and
other chaotic phenomena.

Quantum field theories
========================
Other than these classical continuum field theories, the most widely
known gauge theories are quantum field theories, including quantum
electrodynamics and the Standard Model of elementary particle physics.
The starting point of a quantum field theory is much like that of its
continuum analog: a gauge-covariant action integral that characterizes
"allowable" physical situations according to the principle of least
action. However, continuum and quantum theories differ significantly
in how they handle the excess degrees of freedom represented by gauge
transformations. Continuum theories, and most pedagogical treatments
of the simplest quantum field theories, use a gauge fixing
prescription to reduce the orbit of mathematical configurations that
represent a given physical situation to a smaller orbit related by a
smaller gauge group (the global symmetry group, or perhaps even the
trivial group).

More sophisticated quantum field theories, in particular those that
involve a non-abelian gauge group, break the gauge symmetry within the
techniques of perturbation theory by introducing additional fields
(the Faddeev-Popov ghosts) and counterterms motivated by anomaly
cancellation, in an approach known as BRST quantization. While these
concerns are in one sense highly technical, they are also closely
related to the nature of measurement, the limits on knowledge of a
physical situation, and the interactions between incompletely
specified experimental conditions and incompletely understood physical
theory. The mathematical techniques that have been developed in order
to make gauge theories tractable have found many other applications,
from solid-state physics and crystallography to low-dimensional
topology.

Classical electromagnetism
============================
Historically, the first example of gauge symmetry discovered was
classical electromagnetism. In electrostatics, one can either discuss
the electric field, E, or its corresponding electric potential, 'V'.
Knowledge of one makes it possible to find the other, except that
potentials differing by a constant, V \rightarrow V+C, correspond to
the same electric field. This is because the electric field relates to
'changes' in the potential from one point in space to another, and the
constant 'C' would cancel out when subtracting to find the change in
potential. In terms of vector calculus, the electric field is the
gradient of the potential, \mathbf{E} = -\nabla V. Generalizing from
static electricity to electromagnetism, we have a second potential,
the vector potential A, with
:\begin{align}
\mathbf{E} &= -\nabla V - \frac{\partial \mathbf{A}}{\partial
t}\\
\mathbf{B} &= \nabla \times \mathbf{A}
\end{align}

The general gauge transformations now become not just V \rightarrow
V+C but
:\begin{align}
\mathbf{A} &\rightarrow \mathbf{A} + \nabla f\\
V &\rightarrow V - \frac{\partial f}{\partial t}
\end{align}

where 'f' is any twice differentiable function that depends on
position and time. The fields remain the same under the gauge
transformation, and therefore Maxwell's equations are still satisfied.
That is, Maxwell's equations have a gauge symmetry.

An example: Scalar O(''n'') gauge theory
==========================================
:'The remainder of this section requires some familiarity with
classical or quantum field theory, and the use of Lagrangians.'

:'Definitions in this section: gauge group, gauge field, interaction
Lagrangian, gauge boson.'

The following illustrates how local gauge invariance can be
"motivated" heuristically starting from global symmetry properties,
and how it leads to an interaction between originally non-interacting
fields.

Consider a set of 'n' non-interacting real scalar fields, with equal
masses 'm'. This system is described by an action that is the sum of
the (usual) action for each scalar field \varphi_i

: \mathcal{S} = \int \, \mathrm{d}^4 x \sum_{i=1}^n \left[ \frac{1}{2}
\partial_\mu \varphi_i \partial^\mu \varphi_i - \frac{1}{2}m^2
\varphi_i^2 \right]

The Lagrangian (density) can be compactly written as

:\ \mathcal{L} = \frac{1}{2} (\partial_\mu \Phi)^T \partial^\mu \Phi -
\frac{1}{2}m^2 \Phi^T \Phi

by introducing a vector of fields
:\ \Phi = ( \varphi_1, \varphi_2,\ldots, \varphi_n)^T

The term \partial_\mu is Einstein notation for the partial derivative
of \Phi in each of the four dimensions.

It is now transparent that the Lagrangian is invariant under the
transformation

:\ \Phi \mapsto \Phi' = G \Phi

whenever 'G' is a 'constant' matrix belonging to the 'n'-by-'n'
orthogonal group O('n'). This is seen to preserve the Lagrangian,
since the derivative of \Phi' transforms identically to \Phi and both
quantities appear inside dot products in the Lagrangian (orthogonal
transformations preserve the dot product).

:\ (\partial_\mu \Phi) \mapsto (\partial_\mu \Phi)' = G \partial_\mu
\Phi

This characterizes the 'global' symmetry of this particular
Lagrangian, and the symmetry group is often called the gauge group;
the mathematical term is structure group, especially in the theory of
G-structures. Incidentally, Noether's theorem implies that invariance
under this group of transformations leads to the conservation of the
'currents'
:\ J^{a}_{\mu} = i\partial_\mu \Phi^T T^{a} \Phi

where the 'Ta' matrices are generators of the SO('n') group. There is
one conserved current for every generator.

Now, demanding that this Lagrangian should have 'local'
O('n')-invariance requires that the 'G' matrices (which were earlier
constant) should be allowed to become functions of the space-time
coordinates 'x'.

In this case, the 'G' matrices do not "pass through" the derivatives,
when 'G' = 'G'('x'),

:\ \partial_\mu (G \Phi) \neq G (\partial_\mu \Phi)

The failure of the derivative to commute with "G" introduces an
additional term (in keeping with the product rule), which spoils the
invariance of the Lagrangian. In order to rectify this we define a new
derivative operator such that the derivative of \Phi again transforms
identically with \Phi

:\ (D_\mu \Phi)' = G D_\mu \Phi

This new "derivative" is called a (gauge) covariant derivative and
takes the form

:\ D_\mu = \partial_\mu - i g A_\mu

Where 'g' is called the coupling constant; a quantity defining the
strength of an interaction.
After a simple calculation we can see that the gauge field 'A'('x')
must transform as follows

:\ A'_\mu = G A_\mu G^{-1} - \frac{i}{g} (\partial_\mu G)G^{-1}

The gauge field is an element of the Lie algebra, and can therefore be
expanded as

:\ A_{\mu} = \sum_a A_{\mu}^a T^a

There are therefore as many gauge fields as there are generators of
the Lie algebra.

Finally, we now have a 'locally gauge invariant' Lagrangian

:\ \mathcal{L}_\mathrm{loc} = \frac{1}{2} (D_\mu \Phi)^T D^\mu \Phi
-\frac{1}{2}m^2 \Phi^T \Phi

Pauli uses the term 'gauge transformation of the first type' to mean
the transformation of \Phi, while the compensating transformation in A
is called a 'gauge transformation of the second type'.

The difference between this Lagrangian and the original 'globally
gauge-invariant' Lagrangian is seen to be the interaction Lagrangian

:\ \mathcal{L}_\mathrm{int} = i\frac{g}{2} \Phi^T A_{\mu}^T
\partial^\mu \Phi + i\frac{g}{2} (\partial_\mu \Phi)^T A^{\mu} \Phi -
\frac{g^2}{2} (A_\mu \Phi)^T A^\mu \Phi

This term introduces interactions between the 'n' scalar fields just
as a consequence of the demand for local gauge invariance. However, to
make this interaction physical and not completely arbitrary, the
mediator 'A'('x') needs to propagate in space. That is dealt with in
the next section by adding yet another term,
\mathcal{L}_{\mathrm{gf}}, to the Lagrangian. In the quantized version
of the obtained classical field theory, the quanta of the gauge field
'A'('x') are called gauge bosons. The interpretation of the
interaction Lagrangian in quantum field theory is of scalar bosons
interacting by the exchange of these gauge bosons.

The Yang�Mills Lagrangian for the gauge field
===============================================
The picture of a classical gauge theory developed in the previous
section is almost complete, except for the fact that to define the
covariant derivatives 'D', one needs to know the value of the gauge
field A(x) at all space-time points. Instead of manually specifying
the values of this field, it can be given as the solution to a field
equation. Further requiring that the Lagrangian that generates this
field equation is locally gauge invariant as well, one possible form
for the gauge field Lagrangian is

:\mathcal{L}_\text{gf} = -\frac{1}{2} \operatorname{Tr}\left(F^{\mu
\nu} F_{\mu \nu}\right) = -\frac{1}{4} F^{a \mu \nu} F^a_{\mu \nu}

where the F^a_{\mu \nu} are obtained from potentials A^a_\mu, being
the components of A(x), by

:F_{\mu \nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a +
g\sum_{b,c} f^{abc} A_\mu^b A_\nu^c

and the f^{abc} are the structure constants of the Lie algebra of the
generators of the gauge group. This formulation of the Lagrangian is
called a Yang-Mills action. Other gauge invariant actions also exist
(e.g., nonlinear electrodynamics, Born-Infeld action, Chern-Simons
model, theta term, etc.).

In this Lagrangian term there is no field whose transformation
counterweighs the one of A. Invariance of this term under gauge
transformations is a particular case of 'a priori' classical
(geometrical) symmetry. This symmetry must be restricted in order to
perform quantization, the procedure being denominated gauge fixing,
but even after restriction, gauge transformations may be possible.

The complete Lagrangian for the gauge theory is now

:\mathcal{L} = \mathcal{L}_\text{loc} + \mathcal{L}_\text{gf} =
\mathcal{L}_\text{global} + \mathcal{L}_\text{int} +
\mathcal{L}_\text{gf}

An example: Electrodynamics
=============================
As a simple application of the formalism developed in the previous
sections, consider the case of electrodynamics, with only the electron
field. The bare-bones action that generates the electron field's Dirac
equation is

:\mathcal{S} = \int \bar{\psi}\left(i \hbar c \, \gamma^\mu
\partial_\mu - mc^2\right) \psi \, \mathrm{d}^4 x

The global symmetry for this system is

:\psi \mapsto e^{i \theta} \psi

The gauge group here is U(1), just rotations of the phase angle of the
field, with the particular rotation determined by the constant 'θ'.

"Localising" this symmetry implies the replacement of θ by θ('x'). An
appropriate covariant derivative is then

:D_\mu = \partial_\mu - i \frac{e}{\hbar} A_\mu

Identifying the "charge" 'e' (not to be confused with the mathematical
constant e in the symmetry description) with the usual electric charge
(this is the origin of the usage of the term in gauge theories), and
the gauge field 'A'('x') with the four-vector potential of
electromagnetic field results in an interaction Lagrangian

:\mathcal{L}_\text{int} = \frac{e}{\hbar}\bar{\psi}(x) \gamma^\mu
\psi(x) A_\mu(x) = J^\mu(x) A_\mu(x)

where J^\mu(x) = \frac{e}{\hbar}\bar{\psi}(x) \gamma^\mu \psi(x) is
the electric current four vector in the Dirac field. The gauge
principle is therefore seen to naturally introduce the so-called
minimal coupling of the electromagnetic field to the electron field.

Adding a Lagrangian for the gauge field A_\mu(x) in terms of the field
strength tensor exactly as in electrodynamics, one obtains the
Lagrangian used as the starting point in quantum electrodynamics.

:\mathcal{L}_\text{QED} = \bar{\psi}\left(i\hbar c \, \gamma^\mu D_\mu
- mc^2\right)\psi - \frac{1}{4 \mu_0}F_{\mu\nu}F^{\mu\nu}

Mathematical formalism
======================================================================
Gauge theories are usually discussed in the language of differential
geometry. Mathematically, a 'gauge' is just a choice of a (local)
section of some principal bundle. A gauge transformation is just a
transformation between two such sections.

Although gauge theory is dominated by the study of connections
(primarily because it's mainly studied by high-energy physicists), the
idea of a connection is not central to gauge theory in general. In
fact, a result in general gauge theory shows that affine
representations (i.e., affine modules) of the gauge transformations
can be classified as sections of a jet bundle satisfying certain
properties. There are representations that transform covariantly
pointwise (called by physicists gauge transformations of the first
kind), representations that transform as a connection form (called by
physicists gauge transformations of the second kind, an affine
representation)�and other more general representations, such as the B
field in BF theory. There are more general nonlinear representations
(realizations), but these are extremely complicated. Still, nonlinear
sigma models transform nonlinearly, so there are applications.

If there is a principal bundle 'P' whose base space is space or
spacetime and structure group is a Lie group, then the sections of 'P'
form a principal homogeneous space of the group of gauge
transformations.

Connections (gauge connection) define this principal bundle, yielding
a covariant derivative � in each associated vector bundle. If a local
frame is chosen (a local basis of sections), then this covariant
derivative is represented by the connection form 'A', a Lie
algebra-valued 1-form, which is called the gauge potential in physics.
This is evidently not an intrinsic but a frame-dependent quantity. The
curvature form 'F', a Lie algebra-valued 2-form that is an intrinsic
quantity, is constructed from a connection form by

:\mathbf{F}=\mathrm{d}\mathbf{A}+\mathbf{A}\wedge\mathbf{A}

where d stands for the exterior derivative and \wedge stands for the
wedge product. (\mathbf{A} is an element of the vector space spanned
by the generators T^{a}, and so the components of \mathbf{A} do not
commute with one another. Hence the wedge product
\mathbf{A}\wedge\mathbf{A} does not vanish.)

Infinitesimal gauge transformations form a Lie algebra, which is
characterized by a smooth Lie-algebra-valued scalar, ε. Under such an
infinitesimal gauge transformation,

:\delta_\varepsilon
\mathbf{A}=[\varepsilon,\mathbf{A}]-\mathrm{d}\varepsilon

where [\cdot,\cdot] is the Lie bracket.

One nice thing is that if \delta_\varepsilon X=\varepsilon X, then
\delta_\varepsilon DX=\varepsilon DX where D is the covariant
derivative

:DX\ \stackrel{\mathrm{def}}{=}\ \mathrm{d}X + \mathbf{A}X

Also, \delta_\varepsilon \mathbf{F} = \varepsilon \mathbf{F}, which
means \mathbf{F} transforms covariantly.

Not all gauge transformations can be generated by infinitesimal gauge
transformations in general. An example is when the base manifold is a
compact manifold without boundary such that the homotopy class of
mappings from that manifold to the Lie group is nontrivial. See
instanton for an example.

The 'Yang-Mills action' is now given by

:\frac{1}{4g^2}\int \operatorname{Tr}[*F\wedge F]

where * stands for the Hodge dual and the integral is defined as in
differential geometry.

A quantity which is gauge-invariant (i.e., invariant under gauge
transformations) is the Wilson loop, which is defined over any closed
path, γ, as follows:

:\chi^{(\rho)}\left(\mathcal{P}\left\{e^{\int_\gamma A}\right\}\right)

where � is the character of a complex representation � and \mathcal{P}
represents the path-ordered operator.

The formalism of gauge theory carries over to a general setting. For
example, it is sufficient to ask that a vector bundle have a metric
connection; when one does so, one finds that the metric connection
satisfies the Yang-Mills equations of motion.

Quantization of gauge theories
======================================================================
Gauge theories may be quantized by specialization of methods which are
applicable to any quantum field theory. However, because of the
subtleties imposed by the gauge constraints (see section on
Mathematical formalism, above) there are many technical problems to be
solved which do not arise in other field theories. At the same time,
the richer structure of gauge theories allows simplification of some
computations: for example Ward identities connect different
renormalization constants.

Methods and aims
==================
The first gauge theory quantized was quantum electrodynamics (QED).
The first methods developed for this involved gauge fixing and then
applying canonical quantization. The Gupta-Bleuler method was also
developed to handle this problem. Non-abelian gauge theories are now
handled by a variety of means. Methods for quantization are covered in
the article on quantization.

The main point to quantization is to be able to compute quantum
amplitudes for various processes allowed by the theory. Technically,
they reduce to the computations of certain correlation functions in
the vacuum state. This involves a renormalization of the theory.

When the running coupling of the theory is small enough, then all
required quantities may be computed in perturbation theory.
Quantization schemes intended to simplify such computations (such as
canonical quantization) may be called perturbative quantization
schemes. At present some of these methods lead to the most precise
experimental tests of gauge theories.

However, in most gauge theories, there are many interesting questions
which are non-perturbative. Quantization schemes suited to these
problems (such as lattice gauge theory) may be called non-perturbative
quantization schemes. Precise computations in such schemes often
require supercomputing, and are therefore less well-developed
currently than other schemes.

Anomalies
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Some of the symmetries of the classical theory are then seen not to
hold in the quantum theory; a phenomenon called an anomaly. Among the
most well known are:
*The scale anomaly, which gives rise to a 'running coupling constant'.
In QED this gives rise to the phenomenon of the Landau pole. In
quantum chromodynamics (QCD) this leads to asymptotic freedom.
*The chiral anomaly in either chiral or vector field theories with
fermions. This has close connection with topology through the notion
of instantons. In QCD this anomaly causes the decay of a pion to two
photons.
*The gauge anomaly, which must cancel in any consistent physical
theory. In the electroweak theory this cancellation requires an equal
number of quarks and leptons.

Pure gauge
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A pure gauge is the set of field configurations obtained by a gauge
transformation on the null-field configuration, i.e., a
gauge-transform of zero. So it is a particular "gauge orbit" in the
field configuration's space.

Thus, in the abelian case, where A_\mu (x) \rightarrow A'_\mu(x) =
A_\mu(x)+ \partial_\mu f(x), the pure gauge is just the set of field
configurations A'_\mu(x) = \partial_\mu f(x) for all .

See also
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*Gauge principle
*Aharonov-Bohm effect
*Coulomb gauge
*Electroweak theory
*Gauge covariant derivative
*Gauge fixing
*Gauge gravitation theory
*Gauge group (mathematics)
*Kaluza-Klein theory
*Lorenz gauge
*Quantum chromodynamics
* Gluon field
* Gluon field strength tensor
*Quantum electrodynamics
* Electromagnetic four-potential
* Electromagnetic tensor
*Quantum field theory
*Quantum gauge theory
*Standard Model
*Standard Model (mathematical formulation)
*Symmetry breaking
*Symmetry in physics
*Symmetry in quantum mechanics
*Ward identities
*Yang-Mills theory
*Yang-Mills existence and mass gap
*1964 PRL symmetry breaking papers

Bibliography
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;General readers

* Schumm, Bruce (2004)
'[https://books.google.com/books?id=htJbAf7xA_oC&printsec=frontcover#v=onepa
ge&q=%22gauge%20theory%22&f=false
Deep Down Things]'. Johns Hopkins University Press. Esp. chpt. 8. A
serious attempt by a physicist to explain gauge theory and the
Standard Model with little formal mathematics.

;Texts

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External links
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*
* [http://wiki.math.toronto.edu/DispersiveWiki/index.php/Yang-Mills
Yang-Mills equations on DispersiveWiki]
* [http://www.scholarpedia.org/article/Gauge_theories Gauge theories
on Scholarpedia]

License
=========
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Original Article: http://en.wikipedia.org/wiki/Gauge_theory